Constrained Willmore Tori in the 4--Sphere

TL;DR

This paper classifies constrained Willmore tori in the 4-sphere, showing they are either of finite spectral genus or holomorphic type, and can be explicitly constructed using algebraic geometry and integrable systems techniques.

Contribution

It establishes a classification of constrained Willmore tori in the 4-sphere, linking geometric properties to algebraic and integrable systems methods.

Findings

Constrained Willmore tori are either of finite type or holomorphic type.

All such tori can be explicitly constructed via algebraic geometry.

The proof combines quaternionic holomorphic geometry with integrable systems techniques.

Abstract

We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in the 4-sphere can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin's approach to the study of harmonic tori in the 3-sphere.